Especially in regards to church discipline, what is the responsibility of church leaders to committed regular attenders who are not members?
Their responsibility is certainly less than to those who have voluntarily committed themselves to be members of the church. Because that's the meaning of membership.
Membership means, "I now put myself under the care of the elders and I invite them to hold me accountable as a constituent member of this body." That's the meaning of membership.
If you ask what's the difference between member and regular attender it is that a member can say, "I ascribe to what a member is expected to believe, and I submit myself to the disciplinary procedures of this church to hold me accountable to live a certain way and believe a certain way. And if I go astray I want them to follow through with me." Whereas a regular attender is, for one reason or another, not saying that.
Therefore, the way you would relate to them is by simply having the graciousness to do what you can do.
But you can't do the same thing for a regular attender. You can't excommunicate them. You can't have a formal action of the church by which you forbid them from their membership, and their access to the Lord's Table, and their voting, and their operating in certain roles of teaching, and all of that. They can just still sit on the margins and go about their life.
So I think membership is implicit in the New Testament in the way the New Testament teaches about discipline.